Counting Principle FACTORIAL, PERMUTATION, COMBINATION FACTORIAL, PERMUTATION, COMBINATION

Example: number of possible outcomes for the following events to happen If a die is rolled there are 6 possible outcomes, {1, 2, 3, 4, 5, 6}. If a coin is tossed there are 2 possible outcomes, {H, T}.

H T H T H T H T H T H T If the die is rolled and then the coin is tossed there are 6 * 2 or 12 possible outcomes. 1H 1T 2H 2T 3H 3T 4H 4T 5H 5T 6H 6T S= {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}

Using the fundamental counting principle How many different ways can you form US telephone numbers? Number of possible telephone numbers in the US will be: !

Practice: Find the number of pairs you will get when you roll a pair of dice. S= 36

Warm-up List down all the possible outcomes of a single birth for a married couple planning to have 3 children. Show your complete work with a tree diagram. S={BBB, GBB, BGB, BBG, GGB, GBG, BGG, GGG}

Permutation A permutation of n different elements is an ordering of the element such that one element is first, one is second, one is third and so on….

The difference between Permutation and Combination "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. "The combination to the safe was 472". Now we do care about the order. "724" would not work, nor would "247". It has to be exactly

Permutation In how many distinguishable ways can the letters in BANANA be written? n! n1! n2! n3! = 6! 3! 2! 1! = 6x5x4x3! 3! 2! 1! =60 ways

Let’s say we have 8 people:1: Alice2: Bob3: Charlie4: David5: Eve6: Frank7: George8: HoratioHow many ways can we pick a Gold, Silver, and Bronze medal for “Best friend in the world”? 8x7x6=336 8P38P3 = = 8! (8-3 )! = 8x7x6x 5! 5! = 336

Combinations are easy going. Order doesn’t matter. You can mix it up and it looks the same. Let’s say I’m a cheapskate and can’t afford separate Gold, Silver and Bronze medals. In fact, I can only afford empty tin cans.How many ways can I give 3 tin cans to 8 people? 8C38C3 = = 8! 3!(8-3 )! = 8x7x6x 5! 3! 5! =56 Ways

practice In how many different ways can three letters be chosen from the letters A, B, C, D, E? (the order of the three letters is not important) 5C35C3 = = 5! 3!(5-3 )! = 5x4x 3! 3! 2! =10 Ways

The school board has seven members The board must have three officers: a chairperson, an assistant chairperson, and a secretary. How many different sets of these officers can be formed from this board? How many three-person committees can be formed from this board? Is part (a) asking for a number of permutaions or a number of combinations ? What about part (b)? How are your answers to parts (a) and (b) related?